I want to show that if $(f_{n})_{n}$ converges weakly in $L^{p}(\Omega)$ then $(f_{n})_{n}$ is uniformly bounded in $L^{p}$.
The following is my attempt at proving this:
Assume $f_{n} \rightharpoonup 0$ in $L^{p}(\Omega)$ and let $\varphi \in (L^{p}(\Omega))^{*}$, then by Riesz Representation Theorem there exists a unique $u \in L^{p^{'}}(\Omega)$ such that:
$\langle \varphi, f_{n} \rangle = \int_{\Omega} u f_{n}$ for all $n \in \mathbb{N}$
Moreover, $||u||_{L^{p'}(\Omega)} = ||\varphi||_{(L^{p}(\Omega))^{*}}$
Since $f_{n} \rightharpoonup 0$ it follows that $\langle \varphi, f_{n} \rangle \rightarrow 0$ by characterization of weak convergence. We can also define the linear functional as a linear functional on $L^{p'}(\Omega)$ by $\langle \gamma_{n}, u \rangle := \langle \varphi, f_{n} \rangle$, it follows then that $\langle \gamma_{n},u \rangle \rightarrow 0$ is bounded for any $u$ since every convergent sequence is bounded.
$\therefore$ $\text{sup}_{n} |\langle \gamma_{n},u \rangle| < \infty$ by "Uniform Boundedness Principle" it follows that $\text{sup}_{n}||\gamma_{n}||_{(L^{p'})^{*}} < \infty$.
The result that I want is $\text{sup}_{n}||\gamma_{n}||_{(L^{p'})^{*}} = ||f_{n}||_{L^{p}}$. By Riesz Representation Theorem what I have is $||u||_{L^{p'}} = ||\varphi||_{(L^{p})^{*}}$, as stated above.
Can anyone see how this desired result follows from my argument? Is there a different, more efficient way of getting this result? Is this result unique to $L^{p}$ spaces? Thanks.
Best Answer
Expanding on a comment by David Mitra:
You want to define $\gamma_n\in L_p^{**}$ by $\gamma_n(f^*)=f^*(f_n)$ for $f^*\in L_p^*$. If you do this, then $\gamma_n$ is the image of $f_n$ under the canonical embedding of $L_p$ into its second dual. Since the family $\{\gamma_n\}$ is pointwise bounded on $L_p^*$, it is bounded in norm (by the Uniform Boundedness Principle). And since $\|\gamma_n\|_{L_p^{**}}=\|f_n\|_{L_p}$, the conclusion follows.
By the way, nothing here relies on the underlying space being $L_p$; the proof works the same in every normed space $X$. (The dual space $X^*$ is automatically complete, so the Uniform Boundedness Principle applies to it.)